Strong Laws of Large Numbers for Generalizations of Fréchet Mean Sets

2020/12/23 by Christof Schötz, Schötz, Christof · 4 citations
Economics, Econometrics and Finance · Mathematics · Decision Sciences · #Stochastic processes and financial applications #Statistical Methods and Inference #Probability and Risk Models

paper · pdf · doi:10.48550/arxiv.2012.12762

Abstract

A Fréchet mean of a random variable Y with values in a metric space (\mathcal Q, d) is an element of the metric space that minimizes q ↦ \mathbb E[d(Y,q)²]. This minimizer may be non-unique. We study strong laws of large numbers for sets of generalized Fréchet means. Following generalizations are considered: the minimizers of \mathbb E[d(Y, q)^α] for α> 0, the minimizers of \mathbb E[H(d(Y, q))] for integrals H of non-decreasing functions, and the minimizers of \mathbb E[\mathfrak c(Y, q)] for a quite unrestricted class of cost functions \mathfrak c. We show convergence of empirical versions of these sets in outer limit and in one-sided Hausdorff distance. The derived results require only minimal assumptions.

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• Uniform Convergence of Generalized Conditional Fréchet Means with Applications to Weighted Fréchet Aggregation and Exceedance Set Estimation
• On the Kolmogorov-Feller weak law of large numbers for Frechet mean on non-compact symmetric spaces
• General M-estimators of location on Riemannian manifolds: existence and uniqueness

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